Since a bound \(\phi \)  for \(\mathcal {R}\)  can be chosen such that \(\mathcal {R}^{\prime }(x) \in \partial _\phi \mathcal {R}(x)\)  (see Remark REF ), we can simply search for a classical critical point of \(\mathcal {T}_{\alpha , y^\delta }\)  in order to obatin \(\phi \) -critical points. To achieve this, we apply Newton's method, e.g. [1]}, and we find an initial guess for Newton's method by applying Nesterov accelerated gradient descent [2]} to the starting point \(x_0 = 0\) .
In the following we denote by \(x_\alpha ^\delta \)  a critical point of \(\mathcal {T}_{\alpha , y^\delta }\)  and by \(x_\alpha \)  a critical point of \(\mathcal {T}_{\alpha , y_\mathrm {true}}\) . Here, \(x_\alpha \)  is considered as the limit point for the stability considerations for which we consider the choices \(\alpha \in {10^{-2}, 10^{-3}, 10^{-4}}\) . In order to test for convergence we chosen \(\alpha = \alpha (\delta ) = \delta ^q\)  for \(q \in {1, \frac{3}{2}}\) .
For the convergence simulations we consider as the limit point the signal \(x_\mathrm {true}\)  for the cumulative sum problem, as in this case the solution is unique, and we construct an approximate solution for the inpainting problem by finding a critical point of the function \(\mathcal {T}_{\alpha (\delta ), \delta }\)  for \(\delta = 10^{-16}\)  and we denote this solution by \(x_{\texttt {+}}\) . Implementation details and code are publicly availablehttps://git.uibk.ac.at/c7021101/critical-point-regularization.
